On $\mathcal{D}^+_J$ operator on higher dimensional almost Kähler manifolds

Abstract: In this paper, we define $\mathcal{D}^+_J$ operator that is a generalized $\partial\bar{\partial}$ operator on higher dimensional almost K\"{a}hler manifolds. In terms of $\mathcal{D}^+_J$ operator, we study $\bar{\partial}$-problem in almost K\"{a}hler geometry and the generalized Monge-Amp`{e}re equation on almost K\"{a}hler manifolds. Similarly to the K\"{a}hler case, we obtain $C^\infty$ a priori estimates for the solution of the generalized Monge-Amp`{e}re equation on the almost K\"{a}hler manifold $(M,\omega,J)$ depended only on $\omega$ and $J$. Then as done in K\"{a}hler geometry, we study Calabi conjecture for almost K\"{a}hler manifolds. Finally, we will pose some questions in almost K\"{a}hler geometry.